PREFACE

This volume is an outgrowth of a course intended for first year graduate
students or exceptionally advanced undergraduates in their junior or senior
year. The purpose of the course (taught at Northwestern University in 1956-
1957) was twofold: (a) to provide the necessary elementary background for
all branches of modern mathematics involving " analysis" (which in fact
means everywhere, with the possible exception of logic and pure algebra);
(b) to train the student in the use of the most fundamental mathematical tool
of our time—the axiomatic method (with which he will have had very little
contact, if any at all, during his undergraduate years).

It will be very apparent to the reader that we have everywhere emphasized
the conceptual aspect of every notion, rather than its computational aspect,
which was the main concern of classical analysis; this is true not only of the
text, but also of most of the problems. We have included a rather large number
of problems in order to supplement the text and to indicate further interesting
developments. The problems will at the same time afford the student an
opportunity of testing his grasp of the material presented.

Although this volume includes considerable material generally treated in
more elementary courses (including what is usually called " advanced cal-
culus") the point of view from which this material is considered is completely
different from the treatment it usually receives in these courses. The funda-
mental concepts of function theory and of calculus have been presented within
the framework of a theory which is sufficiently general to reveal the scope,
the power, and the true nature of these concepts far better than it is possible
under the usual restrictions of "classical analysis." It is not necessary to
emphasize the well-known " economy of thought" which results from such a
general treatment; but it may be pointed out that there is a corresponding
" economy of notation," which does away with hordes of indices, much in
the same way as "vector algebra" simplifies classical analytical geometry.
This has also as a consequence the necessity of a strict adherence to axiomatic
methods, with no appeal whatsoever to "geometric intuition," at least in the
formal proofs: a necessity which we have emphasized by deliberately ab-
staining from introducing any diagram in the book. My opinion is that thetudents who